Question

In the figure above, a square is inscribed in a circle. If the area of the square is $36$, find the perimeter of the shaded region.

• A. $6+\frac{3\sqrt{2}}{2}\pi$
• B. $6+3\pi$
• C. $6+3\sqrt{2}\pi$
• D. $36+6\sqrt{2}\pi$
• E. $\frac{9}{2}\pi -9$

Answer 1

Answer: (A)

$6+\frac{3\sqrt{2}}{2}\pi$

As given area of square $=36$
So, length of sides of the square $=\sqrt{36}=6$
As we know when a square is inscribed in a circle, the diagonals are diameters of the circle.

Second, the diagonals of a square meet at right angles.

Third, a diagonal of a square is $\sqrt{2}\times$length of one of the sides of square.
So, the length of the diagonal of square or diameter of the circle$=\sqrt{2}\times 6=6\sqrt{2}$
Radius(r) of the circle $=\frac{diameter\left(D\right)}{2}=\frac{6\sqrt{2}}{2}=3\sqrt{2}$.
As,the diagonals of a square meet at right angles,so it divides perimeter of circle in to four equal part.
So, perimeter of circle $=2\pi r=2\pi 3\sqrt{2}=6\pi \sqrt{2}$
The perimeter of the shaded region$=$length of the one of the sides $+\frac{1}{4}\times$Perimeter of circle.
$=6+\frac{1}{4}\times \pi 6\sqrt{2}$$=6+\pi \frac{3\sqrt{2}}{2}$
Hence, option A is correct.